Skip to main content
SpringerLink
Log in
SpringerLink
Log in

The Kinetic 3D Voronoi Diagram: A Tool for Simulating Environmental Processes

  • Chapter
Advances in 3D Geoinformation Systems

Part of the book series: Lecture Notes in Geoinformation and Cartography ((LNGC))

Abstract

Simulations of environmental processes are usually modelled by partial differential equations that are approximated with numerical methods, based on regular grids. An attractive alternative for simulating a fluid flow is the Free-Lagrange Method (FLM). In this paper, I discuss the use of the FLM—based on the Voronoi diagram (VD)—for the modelling of fluid flow in three dimensions (e.g. the movement of underground water or of pollution plumes in the ocean). Such a technique requires the kinetic three-dimensional VD, which is a VD for which the points are allowed to move freely in space. I present a new algorithm for the movement of points in a three-dimensional VD, and show that it can be relatively easy to implement as it is the extension of a simple two-dimensional algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albers G (1991) Three-dimensional dynamic Voronoi diagrams (in German). Ph.D. thesis, Universität Würzburg, Würzburg, Germany.

    Google Scholar 

  2. Albers G, Guibas LJ, Mitchell JSB, and Roos T (1998) Voronoi diagrams of moving points. International Journal of Computational Geometry and Applications, 8:365–380.

    Article  Google Scholar 

  3. Albers G and Roos T (1992) Voronoi diagrams of moving points in higher dimensional spaces. In Proceedings 3rd Scandinavian Workshop On Algorithm Theory (SWAT’92), volume 621 of Lecture Notes in Computer Science, pages 399–409. Springler-Verlag, Helsinki, Finland.

    Google Scholar 

  4. Anselin L (1999) Interactive techniques and exploratory spatial data analysis. In PA Longley, MF Goodchild, DJ Maguire, and DW Rhind, editors, Geographical Information Systems, pages 253–266. John Wiley & Sons, second edition.

    Google Scholar 

  5. Augenbaum JM (1985) A Lagrangian method for the shallow water equations based on the Voronoi mesh-flows on a rotating sphere. In MJ Fritts, WP Crowley, and HE Trease, editors, Free-Lagrange method, volume 238, pages 54–87. Springer-Verlag, Berlin.

    Chapter  Google Scholar 

  6. Aurenhammer F (1991) Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Computing Surveys, 23(3):345–405.

    Article  Google Scholar 

  7. Bajaj C and Bouma W (1990) Dynamic Voronoi diagrams and Delaunay triangulations. In Proceedings 2nd Annual Canadian Conference on Computational Geometry, pages 273–277. Ottawa, Canada.

    Google Scholar 

  8. Bivand R and Lucas A (2000) Integrating models and geographical information systems. In S Openshaw and RJ Abrahardt, editors, Geocomputation, pages 331–363. Taylor & Francis, London.

    Google Scholar 

  9. Bourke P (1992) Intersection of a line and a sphere (or circle). http://astronomy.swin.edu.au/~pbourke/geometry/sphereline/.

  10. Burrough PA, van Deursen W, and Heuvelink G (1988) Linking spatial process models and GIS: A marriage of convenience or a blossoming partnership? In Proceedings GIS/LIS’ 88, volume 2, pages 598–607. San Antonio, Texas, USA.

    Google Scholar 

  11. Chen J, Zhao R, and Li Z (2004) Voronoi-based k-order neighbour relations for spatial analysis. ISPRS Journal of Photogrammetry & Remote Sensing, 59:60–72.

    Article  Google Scholar 

  12. Cignoni P, Montani C, and Scopigno R (1998) DeWall: A fast divide & conquer Delaunay triangulation algorithm in E d. Computer-Aided Design, 30(5):333–341.

    Article  Google Scholar 

  13. De Fabritiis G and Coveney PV (2003) Dynamical geometry for multiscale dissipative particle dynamics. Computer Physics Communications, 153:209–226.

    Article  Google Scholar 

  14. Devillers O (2002) On deletion in Delaunay triangulations. International Journal of Computational Geometry and Applications, 12(3):193–205.

    Article  Google Scholar 

  15. Edelsbrunner H and Shah NR (1996) Incremental topological flipping works for regular triangulations. Algorithmica, 15:223–241.

    Article  Google Scholar 

  16. Erlebacher G (1985) Finite difference operators on unstructured triangular meshes. In MJ Fritts, WP Crowley, and HE Trease, editors, Free-Lagrange Method, volume 238, pages 21–53. Springer-Verlag, Berlin.

    Google Scholar 

  17. Ferrez JA (2001) Dynamic triangulations for efficient 3D simulation of granular materials. Ph.D. thesis, Département de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland.

    Google Scholar 

  18. Field DA (1986) Implementing Watson’s algorithm in three dimensions. In Proceedings 2nd Annual Symposium on Computational Geometry, volume 246–259. ACM Press, Yorktown Heights, New York, USA.

    Google Scholar 

  19. Freda K (1993) GIS and environment modeling. In MF Goodchild, BO Parks, and LT Steyaert, editors, Environmental modeling with GIS, pages 35–50. Oxford University Press, New York.

    Google Scholar 

  20. Fritts MJ, Crowley WP, and Trease HE (1985) The Free-Langrange method, volume 238. Springler-Verlag, Berlin.

    Google Scholar 

  21. Gahegan M and Lee I (2000) Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams. Computers, Environment and Urban Systems, 24(6):509–537.

    Article  Google Scholar 

  22. Gavrilova ML and Rokne J (2003) Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space. Computer Aided Geometric Design, 20:231–242.

    Google Scholar 

  23. Gold CM (1990) Spatial data structures—the extension from one to two dimensions. In Mapping and Spatial Modelling for Navigation, volume 65, pages 11–39. Springer-Verlag, Berlin, Germany.

    Google Scholar 

  24. Gold CM (1991) Problems with handling spatial data—the Voronoi approach. CISM Journal, 45(1):65–80.

    Google Scholar 

  25. Gold CM (1996) An event-driven approach to spatio-temporal mapping. Geomatica, Journal of the Canadian Institute of Geomatics, 50(4):415–424.

    Google Scholar 

  26. Gold CM, Remmele PR, and Roos T (1995) Voronoi diagrams of line segments made easy. In Proceedings 7th Canadian Conference on Computational Geometry, pages 223–228. Quebec City, Canada.

    Google Scholar 

  27. Gold CM, Remmele PR, and Roos T (1997) Voronoi methods in GIS. In M van Kreveld, J Nievergelt, T Roos, and P Widmayer, editors, Algorithmic Foundations of Geographic Information Systems, volume 1340 of Lecture Notes in Computer Science, pages 21–35. Springer-Verlag.

    Google Scholar 

  28. Guibas L, Karaveles M, and Russel D (2004) A Computational Framework for Handling Motion. In Proceedings 6th Workshop on Algorithm Engineering and Experiments, pages 129–141. New Orleans, USA.

    Google Scholar 

  29. Guibas L and Russel D (2004) An empirical comparison of techniques for updating Delaunay triangulations. In Proceedings 20th Annual Symposium on Computational Geometry, pages 170–179. ACM Press, Brooklyn, New York, USA.

    Google Scholar 

  30. Guibas LJ and Stolfi J (1985) Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transactions on Graphics, 4:74–123.

    Article  Google Scholar 

  31. Imai K, Sumino S, and Imai H (1989) Geometric fitting of two corresponding sets of points. In Proceedings 5th Annual Symposium on Computational Geometry, pages 266–275. ACM Press, Saarbrücken, West Germany.

    Google Scholar 

  32. Joe B (1989) Three-dimensional triangulations from local transformations. SIAM Journal on Scientific and Statistical Computing, 10(4):718–741.

    Article  Google Scholar 

  33. Karavelas MI (2004) A robust and efficient implementation for the segment Voronoi diagram. In International Symposium on Voronoi Diagrams in Science and Engineering, pages 51–62. Tokyo, Japan.

    Google Scholar 

  34. Lawson CL (1986) Properties of n-dimensional triangulations. Computer Aided Geometric Design, 3:231–246.

    Article  Google Scholar 

  35. Ledoux H (2006) Modelling three-dimensional fields in geoscience with the Voronoi diagram and its dual. Ph.D. thesis, School of Computing, University of Glamorgan, Pontypridd, Wales, UK.

    Google Scholar 

  36. Ledoux H, Gold CM, and Baciu G (2005) Flipping to robustly delete a vertex in a Delaunay tetrahedralization. In Proceedings International Conference on Computational Science and its Applications—ICCSA 2005, volume 3480 of Lecture Notes in Computer Science, pages 737–747. Springer-Verlag, Singapore.

    Google Scholar 

  37. Mioc D (2002) The Voronoi spatio-temporal data structure. Ph.D. thesis, Département des Sciences Géomatiques, Université Laval, Québec, Canada.

    Google Scholar 

  38. Mostafavi MA (2001) Development of a global dynamic data structure. Ph.D. thesis, Département des Sciences Géomatiques, Université Laval, Québec City, Canada.

    Google Scholar 

  39. Mostafavi MA (2006) Personal communication.

    Google Scholar 

  40. Mostafavi MA and Gold CM (2004) A global kinetic spatial data structure for a marine simulation. International Journal of Geographical Information Science, 18(3):211–228.

    Article  Google Scholar 

  41. Nativi S, Blumenthal MB, Caron J, Domenico B, Habermann T, Hertzmann D, Ho Y, Raskin R, and Weber J (2004) Differences among the data models used by the geographic information systems and atmospheric science communities. In Proceedings 84th Annual Meeting of the American Meteorological Society. Seattle, USA.

    Google Scholar 

  42. Okabe A, Boots B, Sugihara K, and Chiu SN (2000) Spatial tessellations: Concepts and applications of Voronoi diagrams. John Wiley and Sons, second edition.

    Google Scholar 

  43. Parks BO (1993) The need for integration. In MF Goodchild, BO Parks, and LT Steyaert, editors, Environmental Modeling with GIS, pages 31–34. Oxford University Press, New York.

    Google Scholar 

  44. Raper J (2000) Multidimensional geographic information science.Taylor & Francis, London.

    Google Scholar 

  45. Roos T (1991) Dynamic Voronoi diagrams. Ph.D. thesis, Universität Würzburg, Germany.

    Google Scholar 

  46. Schaller G and Meyer-Hermann M (2004) Kinetic and dynamic Delaunay tetrahedralizations in three dimensions. Computer Physics Communications, 162(1):9–23.

    Article  Google Scholar 

  47. Schirra S (1997) Precision and robustness in geometric computations. In M van Kreveld, J Nievergelt, T Roos, and P Widmayer, editors, Algorithmic Foundations of Geographic Information Systems, volume 1340 of Lecture Notes in Computer Science, pages 255–287. Springer-Verlag, Berlin.

    Google Scholar 

  48. Strang WG and Fix GJ (1973) An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, USA.

    Google Scholar 

  49. Sugihara K and Inagaki H (1995) Why is the 3D Delaunay triangulation difficult to construct? Information Processing Letters, 54:275–280.

    Article  Google Scholar 

  50. Sui DZ and Maggio RC (1999) Integrating GIS with hydrological modeling: Practices, problems, and prospects. Computers, Environment and Urban Systems, 23:33–51.

    Article  Google Scholar 

  51. Watson DF (1981) Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Computer Journal, 24(2):167–172.

    Article  Google Scholar 

  52. Zlatanova S, Abdul Rahman A, and Pilouk M (2002) 3D GIS: Current status and perspectives. In Proceedings Joint Conference on Geo-spatial Theory, Processing and Applications. Ottawa, Canada.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. OTB-section GIS Technology, Delft University of Technology, Jaffalaan 9, 2628 BX, Delft, the Netherlands

    Hugo Ledoux

Authors
  1. Hugo Ledoux
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Jaffalaan 9, 2628 BX, Delft, The Netherlands

    Peter van Oosterom , Sisi Zlatanova , Friso Penninga  & Elfriede M. Fendel , ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ledoux, H. (2008). The Kinetic 3D Voronoi Diagram: A Tool for Simulating Environmental Processes. In: van Oosterom, P., Zlatanova, S., Penninga, F., Fendel, E.M. (eds) Advances in 3D Geoinformation Systems. Lecture Notes in Geoinformation and Cartography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72135-2_20

Download citation

Publish with us

Policies and ethics

Search

Navigation